The value of function (3x + 4)mod 7 for 1 is 0, so let us put the value at 0

1 - - - - - -
0 1 2 3 4 5 6

The value of function (3x + 4)mod 7 for 3 is 6, so let us put the value at 6

1 - - - - - 3
0 1 2 3 4 5 6

The value of function (3x + 4)mod 7 for 8 is 0, but 0 is already occupied, let us put the value(8) at next available space(1)

1 8 - - - - 3
0 1 2 3 4 5 6

The value of function (3x + 4)mod 7 for 10 is 6, but 6 is already occupied, let us put the value(10) at next available space(2)

1 8 10 - - - 3
0 1 2 3 4 5 6

2. In an unweighted, undirected connected graph, the shortest path from a node S to every other node is computed most efficiently, in terms of time complexity by
(A) Dijkstra’s algorithm starting from S.
(B) Warshall’s algorithm
(C) Performing a DFS starting from S.
(D) Performing a BFS starting from S.

Answer(D)

* Time Comlexity of the Dijkstra’s algorithm is O(|V|^2 + E)
* Time Comlexity of the Warshall’s algorithm is O(|V|^3)
* DFS cannot be used for finding shortest paths
* BFS can be used for unweighted graphs. Time Complexity for BFS is O(|E| + |V|)

3. A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is the value of n?
(A) 3
(B) 4
(C) 5
(D) 6

Answer (C)
For an n-ary tree where each node has n children or no children, following relation holds

L = (n-1)*I + 1

Where L is the number of leaf nodes and I is the number of internal nodes.

Let us find out the value of n for the given data.

L = 41 , I = 10
41 = 10*(n-1) + 1
(n-1) = 4
n = 5

4. In the following C function, let n >= m.

int gcd(n,m)
{
if (n%m ==0) return m;
n = n%m;
return gcd(m,n);
}

How many recursive calls are made by this function?
(A) Θ(logn)?
(B) Ω(n)
(C) Θ(loglogn)
(D) Θ(sqrt(n))